My lectures on equivariant cohomology are spinning a bit out of control. The questions and lively discussion, while *always* welcome, have stretched what was meant to be a hand-waving tour through the basics into a three week mini-course (at least, I hope its only three weeks). I’m starting to feel a bit sheepish, since the result I’m trying to get to might not really merit a full month-long preamble.

Last time, I talked about how to define the equivariant cohomology of a space in terms of the cohomology of some big infinite-dimensional space . This is good on a conceptual level, but unless is particularly nice, we will have a bitch of a time computing the cohomology of anything. What we need is a more effective model for the cohomology of .

The idea is to start by pining for the existance of a nice de Rham complex on . We’ll say “Oh, if only it existed, it would look like this, and this…”. Since was only defined by its properties (contractibilty and a free action), this amounts to listing what properties a differential graded algebra should have to correspond to those of . Such DGAs will be called ‘locally-free, acyclic -algebras’.

From there, its a three step process. First, show that *every* such DGA computes the same cohomology. Second, show that there is an (almost) universal locally-free, acyclic -algebra called the ‘Weil algebra’, which is simple enough in structure to make computations effective. Third, show that there exists *any* such DGA which correctly computes the equivariant cohomology (this last step should probably be first, but it isn’t very exciting).